Exponential decay of correlations for Gibbs measures on attractors of Axiom A flows

TL;DR

This paper investigates the decay of correlations in Gibbs measures on Axiom A attractors of flows, establishing conditions for exponential decay or joint integrability of stable and unstable bundles.

Contribution

It proves that codimension one Axiom A attractors with a smooth stable foliation exhibit exponential decay of correlations or have jointly integrable bundles, with implications for flow dynamics.

Findings

Exponential decay of correlations for certain Gibbs measures.

Existence of open sets of flows with exponential mixing and invariance principles.

Growth rate of closed orbits related to topological entropy.

Abstract

In this paper we study the decay of correlations for Gibbs measures associated to codimension one Axiom A attractors for flows. We prove that a codimension one Axiom A attractors whose strong stable foliation is $C^{1+\alpha}$ either have exponential decay of correlations with respect to all Gibbs measures associated to H\"older continuous potentials or their stable and unstable bundles are jointly integrable. As a consequence, there exist $C^1$-open sets of $C^3$-vector fields generating Axiom A flows having attractors so that: (i) mix exponentially with respect to equilibrium states associated with H\"older continuous potentials, (ii) their time-1 maps satisfy an almost sure invariance principle, and (iii) the growth of the number of closed orbits of length $T$ is described by the topological entropy of the attractor.